A difficulty here presented itself, which I must state before proceeding further, as much stress had been laid upon it, and it unquestionably exercised a strong influence upon the subsequent arrangements. It is this: the simple groined vault being formed by the intersection of demi-cylinders, demanded that the space covered by it should be divided into perfect _squares_. Now, the aisles of a church being usually about half the width of the nave, it follows that the groining of both cannot be _square_. If those of the aisles are so, those of the main vault must be about twice as wide as they are long (Fig. 8); while if these are made square, those of the aisles will be twice as long as they are wide (Fig. 9). The first alternative was that most usually adopted north of the Alps, though the second was more frequent in Italy. The difficulty was how to groin these oblong bays. It was not, however, a _new_ difficulty; it had occurred in Roman structures, where it was met by the simple expedient of raising the springing of the narrower vault so high, that its crown was level with that of the wider one. This answered the purpose, but it produced a most unpleasant line of intersection, reducing the vault, in fact, for a portion of its height, to a mere strip of the _arc-doubleau_, and giving a _winding_ intersection for the remainder of the height, as two cylinders of unequal diameter do not intersect in a _plane_. The mathematical solution of the problem would have been to make the section of the narrower vault, an upright _semi-ellipse_; but this does not appear to have been at any period adopted, or, if at all, in exceptional cases only. The pointed arch would have been an approximate expedient, and its introduction has been very ingeniously attributed to this difficulty,--a theory to which I shall have again to allude. Another solution of it would be to make all the arches semi-circles, but to raise up the crown of the vaults of a smaller diameter in a curve to meet the others, thus making it (roughly speaking) a portion of an _annulus_ instead of a _cylinder_. This had one great disadvantage: that it cut off a considerable portion of the space for the clerestory windows; or, if the level of the main vault was raised to obviate this, it became impossible to have a tiebeam to the roof. The system actually adopted in most instances would appear to have been a union of that last named with the Roman mode of stilting the narrow vaults, the difference of height being made up _partly_ by raising its springing, and _partly_ by elevating the crown (Fig. 10). [Illustration: Fig. 10.]